The KMOS Redshift One Spectroscopic Survey ( KROSS ) is an ESO guaranteed time survey of 795Â typical star-forming galaxies in the redshift range z = 0.8 - 1.0 with the KMOS instrument on the VLT . In this paper we present resolved kinematics and star formation rates for 584Â z \sim 1 galaxies . This constitutes the largest near-infrared Integral Field Unit survey of galaxies at z \sim 1 to date . We demonstrate the success of our selection criteria with 90 % of our targets found to be H \alpha emitters , of which 81 % are spatially resolved . The fraction of the resolved KROSS sample with dynamics dominated by ordered rotation is found to be 83 \pm 5 \% . However , when compared with local samples these are turbulent discs with high gas to baryonic mass fractions , \sim 35 \% , and the majority are consistent with being marginally unstable ( Toomre Q \sim 1 ) . There is no strong correlation between galaxy averaged velocity dispersion and the total star formation rate , suggesting that feedback from star formation is not the origin of the elevated turbulence . We postulate that it is the ubiquity of high ( likely molecular ) gas fractions and the associated gravitational instabilities that drive the elevated star-formation rates in these typical z \sim 1 galaxies , leading to the ten-fold enhanced star-formation rate density . Finally , by comparing the gas masses obtained from inverting the star-formation law with the dynamical and stellar masses , we infer an average dark matter to total mass fraction within 2.2 r _ { e } ( 9.5 kpc ) of 65 \pm 12 \% , in agreement with the results from hydrodynamic simulations of galaxy formation .